Tight path, what is it (Ramsey-)good for? Absolutely (almost) nothing!

Abstract

Given a pair of~$k$-uniform hypergraphs~$(G,H)$, the \emph{Ramsey number} of~$(G,H)$, denoted by~$R(G,H)$, is the smallest integer~$n$ such that in every red/blue-colouring of the edges of~$K_n^{(k)}$ there exists a red copy of~$G$ or a blue copy of~$H$. Burr showed that, for any pair of graphs~$(G,H)$, where $G$ is large and connected, the Ramsey number~$R(G,H)$ is bounded below by~$(v(G)-1)(\chi(H)-1)+\sigma(H)$, where~$\sigma(H)$ stands for the minimum size of a colour class over all proper~$\chi(H)$-colourings of~$H$. Together with Erd\H{o}s, he then asked when this lower bound is attained, introducing the notion of Ramsey goodness and its systematic study. We say that~$G$ is \emph{$H$-good} if the Ramsey number of~$(G,H)$ is equal to the general lower bound. Among other results, it was shown by Burr that, for any graph~$H$, every sufficiently long path is~$H$-good. Our goal is to explore the notion of Ramsey goodness in the setting of 3-uniform hypergraphs. Motivated by Burr's result concerning paths and a recent result of Balogh, Clemen, Skokan, and Wagner, we ask: what 3-graphs~$H$ is a (long) tight path good for? We demonstrate that, in stark contrast to the graph case, long tight paths are generally not~$H$-good for various types of 3-graphs~$H$. Even more, we show that the ratio $R(P_n, H)/n$ for a pair~$(P_n, H)$ consisting of a tight path on~$n$ vertices and a 3-graph~$H$ cannot in general be bounded above by \emph{any} function depending only on~$\chi(H)$. We complement these negative results with a positive one, determining the Ramsey number asymptotically for pairs~$(P_n, H)$ when~$H$ belongs to a certain family of hypergraphs.